Problem 128 - Chemistry

Question

Alveoli are the tiny sacs of air in the lungs (see Problem 5.136 ) whose average diameter is \(5.0 \times\) \(10^{-5} \mathrm{~m} .\) Consider an oxygen molecule \(\left(5.3 \times 10^{-26} \mathrm{~kg}\right)\) trapped within a sac. Calculate the uncertainty in the velocity of the oxygen molecule. (Hint: The maximum uncertainty in the position of the molecule is given by the diameter of the sac.)

Step-by-Step Solution

Answer

The uncertainty in the velocity of the oxygen molecule trapped within the alveolar sac in the lungs is equal to or larger than approximately \(0.033 m/s\).

Step 1: Understand the Heisenberg Uncertainty Principle

The Heisenberg's Uncertainty Principle states that you cannot simultaneously know the exact position and momentum (mass times velocity) of a particle. In terms of positioning and velocity, the principle can be represented as: \(\Delta x \cdot \Delta v \geq \frac{h}{4\pi}\) where \(\Delta x\) is the uncertainty in position, \(\Delta v\) is the uncertainty in velocity, and \(h\) is the Planck's constant (\(6.63 \times 10^{-34} Js\)).

Step 2: Use the Given Data to Calculate the Uncertainty in Velocity

We know \(\Delta x\), the uncertainty in position, is the diameter of the alveoli sac \(5.0 \times 10^{-5} m\). Rearrange the Heisenberg Uncertainty Principle equation to solve for \(\Delta v\), we get: \(\Delta v \geq \frac{h}{4\pi \Delta x}\). Then substitute the given \(h = 6.62 \times 10^{-34} Js\) and \(\Delta x = 5.0 \times 10^{-5} m\) to calculate \(\Delta v\).

Step 3: Calculate The Uncertainty in Velocity of The Oxygen Molecule

By substituting the values into the equation \(\Delta v \geq \frac{h}{4\pi \Delta x}\), we find \(\Delta v \geq 0.033 m/s\).

Key Concepts

Alveoli

Alveoli are tiny air sacs within our lungs where a vital exchange of gases occurs. These minuscule structures are the endpoint of your respiratory tree and play a crucial role in your breathing system. They encompass an enormous surface area when all are combined, allowing for vital gas exchange to efficiently take place between the air in the alveoli and the blood in your capillaries.

The primary function of alveoli is to facilitate the movement of oxygen from the inhaled air into the bloodstream, and simultaneously release carbon dioxide from the blood to be exhaled. Alveoli are remarkable for their many attributes:

• Structure: Alveoli are sac-like with thin walls, which maximize the gas exchange surface.
• Size: The average alveolus has a diameter of about \(5.0 \times 10^{-5} \text{ m}\).
• Function: They enable efficient gas exchange due to their extensive surface area and the closeness of blood capillaries.

Understanding alveoli is critical for comprehending how oxygen molecules enter our blood through respiration, making them an important part of studying the movement and uncertainty of molecules, such as in the Heisenberg Uncertainty Principle.

Oxygen molecule

Oxygen molecules are essential for life as we know it. They consist of two oxygen atoms bonded together, commonly referred to as \(O_2\). Each molecule has a mass of approximately \(5.3 \times 10^{-26} \text{ kg}\), making them very tiny but absolutely essential.

These molecules are crucial for cellular respiration, as they act as the primary electron acceptor during the process of converting food into energy in our cells. Here are some key concepts related to the role and behavior of oxygen molecules:

• In Respiration: Oxygen molecules are absorbed by the alveoli and transported through the bloodstream to cells.
• Molecular Movement: The small size of oxygen molecules allows them to move quickly and easily across the thin walls of the alveoli into the blood.
• Statistical Nature: The behavior and movement of gases is often described statistically, particularly when looking at groups of molecules instead of individual ones.

When delving into physical principles like the Heisenberg Uncertainty Principle, oxygen molecules serve as an excellent example to illustrate concepts of molecular movement and uncertainty in small systems.

Planck's constant

Planck's constant is a fundamental constant in physics, playing a crucial role in quantum mechanics. It is denoted by the letter \(h\) and has a value of \(6.63 \times 10^{-34} \text{ Js}\). This constant is incredibly tiny, reflecting the scale at which quantum effects manifest. Understanding Planck’s constant is key to comprehending the Heisenberg Uncertainty Principle and other quantum phenomena.

Here's why Planck’s constant matters:

• Quantum Scale: Planck's constant sets the scale for quantum effects and determines the size of quantum particles and interactions.
• Uncertainty Principle: In Heisenberg's formulation, \(h\) helps quantify the trade-offs in the accuracy of measuring a particle's position and momentum.
• Energy Quanta: Planck's constant is also pivotal in determining the energy levels in atoms, influencing the fundamental theory of quantum mechanics.

By applying Planck's constant in calculations such as the Heisenberg Uncertainty Principle, we gain insight into the limits of precision and how particles like oxygen molecules behave at a microscopic level.